BHV 390: Research Methods

Probability Sampling Techniques

Kimberly Porter Martin, Ph.D.

What is a Population?

DEFINITION: The group to which you want to

generalize your findings.

IN OTHER WORDS:The larger group you are representing with your

sample.ORThe larger group to which your results will apply.

What is a Sample?

DEFINITION A subset of the population being

studied from which data is actually collected.

A good sample accurately represents all kinds of elements/members in proportion to their presence in the population.

Sampling Techniques

Sampling techniques are the processes by which the subset of the population from which you will collect data are chosen.

There are TWO general types of sampling techniques:

1) PROBABILITY SAMPLING

2) NON-PROBABILITY SAMPLING

Probability Sampling

The process of selecting a sample from a population using (statistical) probability theory insuring that

1) each element/member of the population has an equal chance of being included in the sample, and

2) the researcher can estimate the error caused by not collecting data from all elements/members of the population (called “sampling error”).

Frames

DEFINITION

A frame is a list of EVERY element/member of a population.

In order to do probability sampling, you MUST have access to a frame for the population you have chosen. You CANNOT do probability sampling without a frame.

Probability Sampling is ALWAYS superior to Non-probability Sampling.

Probability Sampling is more difficult and time consuming and is not always possible.

Types of Probability Sampling

1.Simple Random Sample (SRS)

2.Systematic Sample with a Random Start (SSRS)

3.Stratified Sample (SS)

4.Multistage Cluster Sample (MCS)

Simple Random Sample

Steps in a Simple Random Sample (SRS)

1. Choose and nominally define a population2. Locate a frame for the population3. Choose a sample size4. Assign a consecutive number to the

elements/members of the frame 5. Obtain a table of random numbers (TRN)6. Count the number of digits in the number assigned to

the LAST member/element of the frame.7. Choose a random starting point in the TRN8. Begin at that point and mark off sets of numbers in

the TRN that have the same number of digits as the last member/element of your frame.

9. Match the numbers you have taken from the TRN to those on your frame

10. The members/elements that match the numbers from the TRN are the members of your sample.

Example of a Simple Random Sample

1. My population is traditional-aged students at ULV.

2. I can get a list of all 1684 traditional-aged students at ULV from the Registrar’s office.

3. I want to collect data from 100 people from this population.

4. I begin at the top of the list (frame) of students and number the students, beginning with 1 and ending with 1684.

Example of a Numbered Frame

1. Mary Aalpoel

2. John Abbinton

3. Oscar Ackerman

.

28. Temecia Kennedy

29. Albert Kostas

.

871.Jose Magana

872. Sarah Martin

968. Josephine Morales

969. Zachary Morton

.

.

.

1000. Bill Zimmerman

• Agnes Zuckerman

Example of a Simple Random Sample

5. I get a statistics book that has a table of random numbers as an appendix.

6. The last person listed in my frame has the number 1684. That number has 4 digits.

7. I choose a random starting point in the TRN.8. I count off 100 sets of four digits beginning

at the random starting point (one set of digits for each person that I want to have in my sample.

Example of Selecting Numbers from a TRN

So far the numbers selected for the sample (those that fall between 0 and 1684) are 0002, 0986, 0872, 1000 and 0028. The process continues until 100 numbers have been randomly selected between 0 and 1684.

Example of a Simple Random Sample

9. I list the 100 numbers from the TRN that fall between 1 and 1684 and match those numbers with those in the frame.

10.The student having the selected numbers will be asked to participate in my study. They will make up my sample of 100.

Sample: 2. John Abbington 968. Josephine

Morales 28. Temecia Kennedy 1000. Bill Zimmerman872. Sarah Martin etc.

Systematic Sample with a Random Start

Steps in a Systematic Sample with a Random Start (SSRS)

1. Choose and nominally define a population

2. Locate a frame for the population

3. Choose a sample size

4. Examine the elements/members in frame for patterns in their status characteristics/attributes. If patterns are present, go to Stratified Sample.

5. Assign a consecutive number to each element in the frame.

6. Determine the sampling interval.

The Sampling Interval In an SSRS you will be systematically selecting

members of your sample by counting off in your frame. The sampling interval tells you how far to count before selecting the next member of your sample. The sampling interval (k) is calculated:

K = P/s

where

P = the population size

(the number of elements in your frame),

and

s = your sample size

Steps in a Systematic Sample with a Random Start (SSRS)

7. Choose a random starting point in your frame.

8. Beginning with the random starting point, count off k element/members of the frame and select the kth element as a member of your sample. Continue, selecting each kth element/member of the frame to be included in your sample.

Example of a Systematic Sample with a Random Start

1. Again, my population is traditional-aged students at ULV.

2. I can get a list of all 1684 traditional-aged students at ULV from the Registrar’s office.

3. I want to collect data from at least 100 people in this population.

4. I begin at the top of the list (frame) of students and number the students, beginning with 1 and ending with 1684 (there are 1684 traditional-aged students registered at this time.

Example of a Numbered Frame

1. Mary Aalpoel

2. John Abbinton

3. Oscar Ackerman

.

28. Temecia Kennedy

29. Albert Kostas

.

871. Jose Magana

872. Sarah Martin

968. Josephine Morales

969. Zachary Morton

.

.

.

1000. Bill Zimmerman

• Agnes Zuckerman

Example of a Systematic Sample with a Random Start

5. I get a statistics book that has a random numbers table as an appendix.

6. The last person listed in my frame has the number 1684. That number has 4 digits.

7. I choose a random starting point in the TRN.8. Beginning at the random starting point, I mark

off the numbers in the TRN by fours until I reach the first set of four digits that falls between 1 and 1684. That is my random starting point for beginning the count off in my frame.

Example of Selecting a Starting Point for SRSS from a TRN

Here I have started at a random point in a TRN and have marked off sets of four numbers until I reached the first number that is between 1 and 1684. That number is 0138. I will therefore start counting off in my frame with student number 138.

Example of Selecting Elements/Members for an SRSS

Sample Beginning with 138 as a starting point, I count off

16, ending with 153. I select member/element number 153 to be in my sample. I add 16 to 153 to arrive at member/element 169 in my frame, and select number 169 to be in my sample. I continue to select every 16th member/element in the frame to be in my sample until I have reached the end of my frame. When I reach the end of the list, I continue my count at the beginning of the list until I have reached 138 again. That will give me a sample size of 105 students.

Example of SRSS Sample Members Selected from a Numbered Frame with k = 16 and a

random starting point of 138153. Albert Kirby169. Sally James185. Temecia Jamison201. Derek Jones217. Susan Johnston233. John Maloney249. Sarah Martin...937. Josephine Solana953. Jesus Soledad.

969. Henry Suzuki...1615. Jesse Wirth1647. Martha Zalm1663. Todd Zimmerman1679. Agnes Zuckerman1. 1. Sandra Aalpoel17. Joan Anderson33. Ian AtchisonEtc until 138 is reached again.

Stratified Sample

Steps in a Stratified Sample

1. Choose and nominally define a population2. Locate a frame for the population3. Choose a sample size4. Examine the elements/members in frame for patterns in their status

characteristics or attributes. 5. Reorganize the elements by grouping all those with the same

status characteristics together on the list (stratify them). The rest of the procedure is the same as it is for an SRSS.

6. Assign a consecutive number to each of the elements/members of the frame as they appear in their new order.

7. Determine the sampling interval, k.8. Choose a random starting point in your frame.9. Beginning with the random starting point, count off k

element/members of the frame and select the kth element as a member of your sample. Continue, selecting each kth element/member of the frame to be included in your sample.

Finding Status Patterns in a Frame

Lets imagine that your population is the personnel in a particular military platoon that is made up of 10 squads of 10 soldiers, 7 of whom are privates, 2 of whom are sergeants and 1 of whom is a lieutenant. You are given a frame (a list of all personnel in the platoon) by the commanding officer. The frame lists the personnel by squad beginning with the lowest ranking soldier and ending with the highest ranking soldier. The following slide shows what the frame would look like.

Sample Frame for Fictional Platoon1. Private2. Private3. Private4. Private5. Private6. Private7. Private8. Sergeant9. Sergeant10.Lieutenant11.Private12.Private13.Private

14. Private15. Private16. Private17. Private18. Sergeant19. Sergeant20. Lieutenant21. Private22. Private23. Private24. Private25. Private26. Private27. Private

28. Sergeant29. Sergeant30. Lieutenant31. Private32. Private33. Private

1. Private34. Sergeant35. Sergeant36. Lieutenant37. Private38. Private39. PrivateEtc. To 100 soldiers

Example of a Stratified Sample1. My population is soldiers in a single platoon.2. I can get a list of all 100 soldiers in this platoon from

the commanding officer.3. I want to do in-depth interviews with 10 soldiers in the

platoon.4. I check the frame and find there is a clear pattern in

the way that soldiers are listed. They are listed by rank. Since I want to interview 10 soldiers and I have a population of 100, my sampling interval for this group would be 10. If I just sample as I would in a SSRS, I would take every 10th soldier. However, choosing every 10th soldier would give me 10 lieutenants, no sergeants and no privates. This is not representative of the group as a whole. I want to have opinions that reflect the whole platoon, and I want to do interviews with people of different ranks in the proportions in which they exist in the platoon.

Straight SRSS Sampling Results1. Private2. Private3. Private4. Private5. Private6. Private7. Private8. Sergeant9. Sergeant10.Lieutenant11.Private12.Private13.Private

14. Private15. Private16. Private17. Private18. Sergeant19. Sergeant20. Lieutenant21. Private22. Private23. Private24. Private25. Private26. Private27. Private

28. Sergeant29. Sergeant30. Lieutenant31. Private32. Private33. Private34. Private35. Private36. Private37. Private38. Sergeant39. Sergeant40.LieutenantEtc. To 100 soldiers

Example of a Stratified Sample

5. I reorganize the elements in the frame by grouping all those with the same status characteristics together on the list (stratify them). This means all privates go together, all sergeants go together and all lieutenants go together in the list.

6. I then assign a consecutive number to each of the soldiers listed in the frame as they appear in their new order.

7. I determine my sampling interval by dividing my population size (100) by the my sample size (10). k = 100/10 = 10.

8. I locate a TRN and choose a random starting point using the same techniques that I used in the SRSS. In this case, my population size is 100 so I need to use three digits to be sure that every soldier has an equal chance of being included.

Example of Selecting a Starting Point for SRSS from a TRN

Here I have started at a random point in a TRN and have marked off sets of three numbers until I reached the first number that is between 1 and 100. That number is 050. I will therefore start counting off in my frame with soldier number 50.

Example of a Stratified Sample

7.Beginning with the random starting point, I count off 10 soldiers and select the 10th as a member of my sample. I continue, selecting each 10th soldier as I go through the frame. When I get to 100, I continue uninterrupted in my count until I have reached. The resultant sample will have exactly the proportion of privates, sergeants and lieutenants as are in the platoon: 7 privates, 2 sergeants and 1 lieutenant

SRSS Sampling Resultswith Stratification

50.Private51.Private52.Private53.Private54.Private55.Private56.Private57.Private58.Private59.Private60.Private61.Private62.Private

63. Private64. Private65. Private66. Private67. Private68. Private69. Private70. Private71. Lieutenant72. Lieutenant73. Lieutenant74. Lieutenant75. Lieutenant76. Lieutenant

77.Lieutenant78.Lieutenant79. Lieutenant80. Lieutenant81. Sergeant82. Sergeant83. Sergeant84. Sergeant85. Sergeant86. Sergeant87. Sergeant88. Sergeant89. SergeantEtc until 10 selected

59. Private 9. Private

69. Private 19. Private

79. Lieutenant 29. Private

89. Sergeant 39. Private

99. Sergeant 49. Private

Example of SS Sample Members Selected from a Numbered Frame with P = 100, k = 10 and a random starting point of 50

The result is a sample of 10 with 10% Lieutenants, 20% Sergeants and 70% Privates

Multistage Cluster Sampling

What is Multistage Cluster Sampling?

Multistage cluster sampling is a technique for:

when you do not have a frame that lists all elements of a population,

OR

when numbers of individual elements in your population are too numerous to sample easily.

AND

you can obtain frames for groups of population elements/members.

Steps in Multistage Cluster Sampling

1. Choose and nominally define a population2. Choose a sample size3. Identify groupings of the elements that make up your population. 4. Obtain a frame for the groups of elements.5. Randomly sample the groups using SRS or SSRS.6. Obtain a frame for the individual elements within each group selected during step 5.7. Randomly sample the individuals in the groups selected in step 5 using SRS or SSRS.

Example of MultiStage Cluster Sampling

1. My population is all the people living in the city of La Verne. There is no frame for all people currently living in La Verne.

2. I want to survey 100 residents of La Verne about the City Council’s latest action.

3. I can get a list of all the 1750 streets in La Verne and also for all mailing addresses (representing households) in the city. These groupings of residents will help me use probability sampling to select 100 residents. I will select 25 streets in the city, and then four households from each of the 25 streets.

Example of MultiStage Cluster Sampling

4. I get a frame that includes all streets in La Verne.

5. I use Simple Random Sampling techniques to select 25 Streets in La Verne.

Example of MultiStage Cluster Sampling

There are 1750 streets in La Verne. I will select 25 streets. I begin with a random starting point in a TRN and, because the total number of streets has four digits, I will mark of sets of four digits in the TRN. The following numbers identify streets that will be included in the sample of streets: 910, 850, 505, 50, 1102, 1209, 1092, 750, 40, 1712, and so forth until 25 streets have been selected.

Example of MultiStage Cluster Sampling

6. I then want to select 4 addresses for

each of the 25 streets selected in Step 5. Street number 910 is Pine View, and has 26 addresses on it. I use SRS techniques to select 4 addresses on Pine View. One adult will be surveyed from each of those addresses.

Example of MultiStage Cluster Sampling

There are 26 addresses on Pine View Street. I will select four addresses. I begin with a random starting point in a TRN and, because the total number of addresses has two digits, I will mark of sets of two digits in the TRN. The following numbers identify addresses that will be included in the sample: 2, 9, 17, 19.

Example of Multistage Cluster Sampling

7. I will repeat Step 6. for each of the 25 streets selected in Step 5. At the end of this process, I should have 100 addresses randomly selected, four from each of 25 randomly selected streets.

8. I will visit each of these addresses and interview the first adult I am able to talk with from each household.

Study GuidePopulation

Sample

Probability sampling

Non-probability sampling

Element

Frame

Table of random numbers

Nominal definition

Number of digits in population size

Simple random sample

Random starting point

Systematic sample with a random start

Patterns of status characteristics

Stratified sample

Multistage cluster sample

Populations clusters